Laurent多项式空间、$\mathbb{P}^1$ -orbifolds空间和可积族
The spaces of Laurent polynomials, $\mathbb{P}^1$-orbifolds, and integrable hierarchies
摘要：
设$M_{k,m}$为单变量Laurent多项式空间$x^k + t_1 x^{k-1}+... t_{k+m}x^{-m},$，其中$k,m\geq 1$为固定整数，$t_{k+m}\neq 0$为固定整数.根据B. Dubrovin \cite{D}、$M_{k,m}$可以配备半简单的Frobenius结构。本文证明了$M_{k,m}$（由A. Givental）满足Hirota二次方程（简称HQE）。假设$\mathcal{C}_{k,m}$是从$\mathbb{P}^1$获得的眶褶，通过切割围绕PDMATHAMV和PDMATHAMW的小圆盘PDMATHAMT和$D_2\simeq\{|z^{-1}|\leq ε\}$，并以明显的方式将眶褶$D_1/\mathbb{Z}_k$和$D_2/\mathbb{Z}_m$粘回。证明了$\mathcal{C}_{k,m}$的轨道量子上同调与$M_{k,m}$作为Frobenius流形是一致的.模一些尚未被澄清的细节，这意味着后代（分别祖先）潜在的$M_{k,m}$是一个生成函数的后代（分别祖先）orbifold格罗莫夫-威滕不变量的$\mathcal{C}_{k,m}$。我们的HQE与G. PDMATHAMBMZ中的Carlet。因此，我们的HQE描述了这个层次的τ-函数，我们期望扩展的双阶Toda层次支配$\mathcal{C}_{k,m}.$的Gromov-Witten理论
英文摘要：
Let $M_{k,m}$ be the space of Laurent polynomials in one variable $x^k + t_1 x^{k-1}+... t_{k+m}x^{-m},$ where $k,m\geq 1$ are fixed integers and $t_{k+m}\neq 0$. According to B. Dubrovin \cite{D}, $M_{k,m}$ can be equipped with a semi-simple Frobenius structure. In this paper we prove that the corresponding descendant and ancestor potentials of $M_{k,m}$ (defined by A. Givental) satisfy Hirota quadratic equations (HQE for short). Let $\mathcal{C}_{k,m}$ be the orbifold obtained from $\mathbb{P}^1$ by cutting small discs $D_1\simeq \{|z|\leq ε\}$ and $D_2\simeq\{|z^{-1}|\leq ε\}$ around $z=0$ and $z=\infty$ and gluing back the orbifolds $D_1/\mathbb{Z}_k$ and $D_2/\mathbb{Z}_m$ in the obvious way. We show that the orbifold quantum cohomology of $\mathcal{C}_{k,m}$ coincides with $M_{k,m}$ as Frobenius manifolds. Modulo some yet-to-be-clarified details, this implies that the descendant (respectively the ancestor) potential of $M_{k,m}$ is a generating function for the descendant (respectively ancestor) orbifold Gromov--Witten invariants of $\mathcal{C}_{k,m}$. There is a certain similarity between our HQE and the Lax operators of the Extended bi-graded Toda hierarchy, introduced by G. Carlet in \cite{car}. Therefore, it is plausible that our HQE characterize the tau-functions of this hierarchy and we expect that the Extended bi-graded Toda hierarchy governs the Gromov--Witten theory of $\mathcal{C}_{k,m}.$
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